Recursively enumerable language

About: Recursively enumerable language is a research topic. Over the lifetime, 1508 publications have been published within this topic receiving 32382 citations.

Recursively enumerable extensions of R1 by finite functions

Abstract: Let R1 be the total recursive functions from IN to IN, and Fin the set of all partial functions from IN to IN having a finite initial segment of IN as domain Motivated by earlier studies on "simulation-universal automata" (BUCHBERGER&MENZEL 77/ MAIER, MENZEL&SPERSCHNEIDER 82/ MENZEL &SPERSCHNEIDER 82) we ask what it means that R1UF is recursively enumerable (re), for a subfamily F of Fin We show that each such family F is, in a certain sense, very rich A (simple) corollary is that it must be dense in Fin wrt the usual product topology on Fin As a consequence one obtains simple but useful necessary conditions on F to make R1UF re We also consider the class Ext(R1) :={F⊑Fin | R1UF re} as a whole It is also quite rich in structure (eg if viewed as an upper semi-lattice wrt union of families of functions) A sufficient criterion for F to be in Ext(R1) provides examples of families "naturally in Ext(R1)", thus demonstrating that richness On the other hand, there are families which belong to Ext(R1) in a more nonstandard way Main results: There is an F in Ext(R1) which is itself not re For F in Ext(R1), F is re iff F is "effectively dense in Fin", in some appropriate sense

Prescribed Learning of R.E. Classes

Abstract: This work extends studies of Angluin, Lange and Zeugmann on the dependence of learning on the hypotheses space chosen for the class. In subsequent investigations, uniformly recursively enumerable hypotheses spaces have been considered. In the present work, the following four types of learning are distinguished: class-comprising (where the learner can choose a uniformly recursively enumerable superclass as hypotheses space), class-preserving (where the learner has to choose a uniformly recursively enumerable hypotheses space of the same class), prescribed (where there must be a learner for every uniformly recursively enumerable hypotheses space of the same class) and uniform (like prescribed, but the learner has to be synthesized effectively from an index of the hypothesis space). While for explanatory learning, these four types of learnability coincide, some or all are different for other learning criteria. For example, for conservative learning, all four types are different. Several results are obtained for vacillatory and behaviourally correct learning; three of the four types can be separated, however the relation between prescribed and uniform learning remains open. It is also shown that every (not necessarily uniformly recursively enumerable) behaviourally correct learnable class has a prudent learner, that is, a learner using a hypotheses space such that it learns every set in the hypotheses space. Moreover the prudent learner can be effectively built from any learner for the class.

The Lambek Calculus with Iteration: Two Variants

Abstract: Formulae of the Lambek calculus are constructed using three binary connectives, multiplication and two divisions. We extend it using a unary connective, positive Kleene iteration. For this new operation, following its natural interpretation, we present two lines of calculi. The first one is a fragment of infinitary action logic and includes an omega-rule for introducing iteration to the antecedent. We also consider a version with infinite (but finitely branching) derivations and prove equivalence of these two versions. In Kleene algebras, this line of calculi corresponds to the *-continuous case. For the second line, we restrict our infinite derivations to cyclic (regular) ones. We show that this system is equivalent to a variant of action logic that corresponds to general residuated Kleene algebras, not necessarily *-continuous. Finally, we show that, in contrast with the case without division operations (considered by Kozen), the first system is strictly stronger than the second one. To prove this, we use a complexity argument. Namely, we show, using methods of Buszkowski and Palka, that the first system is \(\varPi _1^0\)-hard, and therefore is not recursively enumerable and cannot be described by a calculus with finite derivations.

Automorphisms of the Lattice of Recursively Enumerable Vector Spaces

Abstract: Recursively enumerable (r.e,) algebraic structures have received attention because of their recursion theoretic depth and richness. Although the development of such theories is somewhat analoguous to the development of the theory of r.e. sets, the former is not reducible to or a corollary of the latter. I n fact, on more than one occasion the development of such theories has increased our insight into the theory of r.e. sets. The initial works in this area are due to FROHLICH and SHEPHERDSON [6] and Rasm [14]. The more recent works on vector space structure are due to DEKKER [4, 51, CROSSLEY and NERODE [3], METAKIDES and NERODE [13], REMMEL [El , RETZLAFF [19], and the author [8]. An excellent motivational reference is METAKIDES and NERODE [El. I n this paper we confine ourselves with vector space structure and study the lattice of r.e. vector spaces and ith automorphisms. Let V , denote a countably infinite dimensional, f d y effwtive vector space over a countable recursive field 3. By fully effective we mean that 8, under a fixed Godel numbering has the following properties : 0) operations of vector addition and scalar multiplication on V, are presented by partial recursive functions on the Godel numbers of 8,. (ii) V , has a depedence algorithm, i.e., there is a uniform effective procedure which applied to any TL vectors of V , determines whether or not they are linearly independent.